Reconstructive Phase Transitions

The following sections are included:

• FOREWORD

• CONTENTS

The following sections are included:

• Introduction

• Basic concepts of the Landau theory of phase transitions
  • Aim of the Landau theory
  • An introductory example
    • Crystallographic description and assumptions
    • Symmetry constraints: Form of the free-energy
    • The order parameter
    • Stable state below T_c and physical anomalies at T_c
    • Symmetry of the order-parameter
    • Degeneracy of the low symmetry phase
  • Generalization of the preceding example: Formulation of the Landau theory
    • Summary of the results obtained in the introductory example
    • Description of a second-order phase transition
    • The transition order-parameter
    • Stable states and symmetry in the vicinity of T_c
    • Summary of the Landau scheme for the study of a phase transition
  • Secondary order-parameters
    • Primary and secondary order-parameters
    • Return to the introductory example
    • Influence of the secondary order-parameter on the properties of the system
    • Physical anomalies related to secondary order-parameters
    • General properties of secondary order-parameters

• The angular and radial problems in the Landau theory of structural transitions
  • Introduction
  • The angular problem and its solutions
    • Image of G₀ in the order-parameter representation
    • Use of the image-group in the application of the Landau theory
    • Integrity bases
    • Isotropy groups and other simplifying mathematical concepts
    • Illustration of the preceding concepts
    • Minima of F and maximal isotropy-groups
    • Minima of F and decomposition into orbits of the integrity basis of I₀
    • Determination of the low-symmetry space groups
  • The radial problem and its solutions
    • Arbitrariness of the phenomenological models: Specific form of the Landau Potential
    • Solutions of the radial problem for one-dimensional order-parameter expansions
    • Phase diagrams for two-component order-parameter expansions
    • Phase diagrams for three-component order-parameter expansions
    • Phase diagrams for image-groups corresponding to an irreducible order-parameter of dimension n≥4
    • Summary. Outline of a theory of phase diagrams. First-order phase transitions

• Symmetry predicted first-order phase transitions
  • Introduction
  • The Landau condition
  • “Lock-in” transitions and the Lifshitz criterion
  • Phase transitions associated with reducible order-parameters
    • General features of the phase diagrams associated with a reducible order-parameter
    • Biquadratic coupling between two order-parameters
    • “Improper” coupling between two order-parameters
    • “Pseudo-proper” coupling between two order-parameters
  • Isostructural and antiisostructural phase transitions
  • First-order phase transitions and stable fixed points

• Theory of singularities and phase transitions
  • Introduction
  • Structural stability
    • Some definitions and concepts of the theory of singularities
    • Splitting lemma. Diffeomorphisms
  • Normal form of a singularity
    • Additional terminology
    • Normal forms: in search of the Landau potential
    • Local nature of the observable singularities
  • Restructuring of phase diagrams. Topological metamorphosis
    • External variables and observable singularities
    • Theory of the restructuring of phase diagrams
    • Topological metamorphosis

• References

The following sections are included:

• Introduction

• Introductory examples to the density-wave approach
  • A first example of displacive reconstructive phase transition
    • Crystallographic description of the transition mechanism
    • Density-wave description of the phase transition
    • Thermodynamic potential and phase diagram
    • Temperature dependence of the order-parameter, and of the critical displacements
    • Other topologies of the phase diagram
  • A second example of displacive reconstructive phase transition
    • Three possible phase transition mechanisms
    • Density-wave approach to the C_4v-C_6v phase transition
  • An example of order-disorder reconstructive phase transition
    • Mechanism of the ordering transition
    • Density-wave description of the reconstructive transition
    • Symmetry of the order-parameter and phase diagram
  • An example of periodic-aperiodic reconstructive phase transition
    • Crystallographic description of the tetragonal-octagonal transformation mechanism
    • Density-wave description of the C_4v–C_8v mechanism

• Generalization of the preceding examples: Density-wave theory of reconstructive phase transitions
  • Differences in the phenomenological descriptions of displacive and order-disorder reconstructive phase transitions
  • Formulation of the density-wave theory, and its connection with the Landau theory
    • Lattice function, probability density, order-parameter
    • Order-parameter space for reconstructive phase transitions
    • Thermodynamic potential, equations of state, stable states

• Specific properties of reconstructive phase transitions
  • Orientational and antiphase domains
  • Temperature dependence of the order-parameter and related physical quantities
  • Specificity of the phase diagrams involving reconstructive transitions
  • Phonon spectra and reconstructive phase transitions

• Applicability of the density-wave theory of reconstructive phase transitions
  • Summary: Specificity of reconstructive phase transitions
  • Parent phase and initial structure
  • Limit (non-Landau) states and limit (Landau) states
  • Intermediate nature of the mechanism in reconstructive transitions
  • Degree of discontinuity of first-order phase transition

• References

The following sections are included:

• Introduction

• The β – ω transition
  • The transition mechanism
  • Direction of the order-parameter in the ε-space
  • Periodicity of the order-parameter magnitude
  • Theoretical phase diagram for the β – ω transition
  • Discussion

• The BCC-HCP transition: The role of macroscopic deformation
  • Burgers mechanism
  • Burgers mechanism reformulated
  • Periodic dependence of the order-parameter magnitude
  • Alternative description of the BCC-HCP transition: The HCP parent phase
  • Thermodynamics of the BCC-HCP transition: Improper symmetry breaking strains

• The BCC-FCC transition
  • The Bain deformation
  • The Bain deformation revised
  • Phase diagram associated with the BCC-FCC transition
  • The BCC-FCC phase diagram

• Landau and limit phases in the phase diagrams of actinides
  • Polymorphism of the actinide group
  • Lighter actinides: Full realization of the Bain deformation
  • Heavier actinides: A variant of Burgers mechanism
  • The transitions mechanisms in uranium, neptunium and plutonium
    • The formation of β-uranium
    • The Landau β- and α -phases of neptunium
    • The idiosyncratic γ, α and β Structures of plutonium
  • Packing of “ovaloid” atoms in actinide structures
  • The charge density waves in α-uranium

• Reconstructive phase transitions in inorganic crystals
  • Breaking of the group-subgroup relationship between the phase-symmetries in complex crystals
  • Silver iodide and copper bromide
  • Beryllium oxide
  • The reconstructive NaCl to CsCl transition in alkali halides
  • The monoclinic-orthorhombic reconstructive transition in K₂ZnBr₄ and K₂CoBr₄

• The martensitic transformations in lithium and sodium
  • Experimental features of the martensitic transformations in Na and Li
  • Phase diagrams of sodium and lithium
  • Phonons and martensitic transformations

• References

The following sections are included:

• Introduction

• The FCC-HCP reconstructive transition
  • Crystallographic description and transition order-parameter
  • Thermodynamic description: The phase diagrams of helium and praseodymium
  • The phonon spectrum of cobalt
  • Stacking faults, antiphase domains, and incomplete ordering

• Polytypic structures in zinc sulfide and barium titanate
  • The wurtzite-sphalerite transition in ZnS
  • The cubic-hexagonal transition in BaTiO₃

• The graphite-diamond reconstructive transition
  • Degree of occupancy of the disordered polytype cell, and symmetry of ordered cubic structures
  • Crystallogeometrical properties of hexagonal graphite and cubic diamond
  • The graphite-diamond transition mechanism
  • Changes in the chemical bonds at the graphite-diamond transition: Physical nature of the order-parameter

• Theory of ice structures
  • The phase diagram of ice: The parent I₀ structure
  • Ordering of the oxygen atoms: The transition mechanisms to the I_h, I_c, ice III and ice VI phases
  • Orientational and positional ordering of the hydrogen atoms: The structures of ice VII, ice VIII, ice IX and ice XI
  • Coupled ordering and displacive mechanisms for the formation of ice II
  • The phase diagram of ice: Connection between pressure, densities, and the fractional concentrations of I₀

• Reconstructive transitions in SiO₂
  • The phase diagram of SiO₂
  • The BCC-stishovite transformation
  • The BCC-tridymite transformation
  • The BCC-β-cristobalite, and β-α cristobalite transformations
  • The BCC-coesite transformation

• Ordering-type and displacive-type mechanisms in lanthanides
  • Different approaches to lanthanide structures
  • Classification of lanthanide structures

• Reconstructive phase transitions in a cryogenic crystal: Nitrogen

• References

The following sections are included:

• Introduction

• The crystal-quasicrystal transition mechanism
  • Symmetrization process and local icosahedral order
  • Relationship between crystal and quasicrystal structures in AlMnSi and AlLiCu
  • Breaking of the translational crystal symmetry: Long-range quasicrystalline order in AlFeSi
    • The incommensurate wave-vector associated with the crystal-quasicrystal transition
    • Deformation of the system of stationary density-waves. Atomic positions in the icosahedral structure
    • Symmetry of the order-parameter. Phase diagram

• Density-wave theory of the FCC-icosahedral transitions in FeTi₂(NiTi₂) alloys
  • Orientational relationship between the FCC and icosahedral structures in FeTi₂
  • Basic system of density waves associated with the inhomogeneous deformation mechanism
  • The critical transition wave-vector. Symmetrization process
  • Atomic positions in the icosahedral FeTi₂ structure
  • The phase diagram

• The rhombohedral-icosahedral transition in AlCuFe alloys
  • Reversible transition between microcrystalline-rhombohedral and ideal icosahedral phases
  • Orientational relationship between the rhombohedral and icosahedral structures
  • Density-wave description of the inhomogeneous deformation mechanism: Primary and secondary order-parameters
  • Incommensurate modulation of the rhombohedral aggregates and lock-in icosahedral phase

• Local structural mechanism for crystal-quasicrystal transitions: The T-lattice
  • Connection between the density-wave and cut-and-projection descriptions. One-dimensional example of T-lattice
  • Two-dimensional example of T-lattice: The tetragonal-to-octagonal transition in MnSi alloys
  • Fusion of sites. Diffusion. Disorder
  • Return to the density-wave description

• Domain structure and microcrystalline states in quasicrystals
  • The T-lattice and the domain structure of quasicrystals
  • The Al₅CuLi₃-Al₆CuLi₃ transition
  • The Al₄Mn-Al₇₈Mn₂₂ (decagonal) transition

• References

The following sections are included:

• APPENDIX 1 GROUP-THEORETICAL ASPECTS OF THE PHENOMENOLOGICAL THEORY OF STRUCTURAL PHASE TRANSITIONS
  • Crystallographic space-groups
    • Geometrical elements composing a space-group
    • Bravais lattice and point-group of a crystal
    • Space-group of a crystal. Unit-cell
  • Irreducible representations of the space-groups
    • Irreducible representations of a group G
    • Irreducible representation of the Translation group. Brillouin-zone. Star of . Small representations of
    • Construction of the irreducible representations of a space group
  • Group theoretical properties used in the formulation of the Landau theory
    • Decomposition of into a sum of irreducible functions
    • Invariant polynomials
    • Reduction of an invariant space into irreducible spaces
    • n-th power of a representation
  • References

• Subject Index